We can consider \(J^{\alpha }, J_{\alpha }\)  for \(v\in L^1(0,T)\) , but
for the moment, we consider them for \(v \in L^2(0,T)\) .
There are many works on the Riemann-Liouville time-fractional integral
operator \(J^{\alpha }\) , and we here refer only to
Gorenflo and Vessella [1]} as monograph, and Gorenflo and
Yamamoto [2]} for an operator theoretical approach.
